Theorem vccl 28826

Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vciOLD.1	⊢ 𝐺 = (1st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vccl	⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Proof of Theorem vccl

Step	Hyp	Ref	Expression
1		vciOLD.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . 3 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . 3 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vcsm 28825	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
5		fovrn 7420	. 2 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
6	4, 5	syl3an1 1161	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   × cxp 5578  ran crn 5581  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  ℂcc 10800  CVec_OLDcvc 28821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805  df-vc 28822

This theorem is referenced by:  vc0  28837  vcm  28839  nvscl  28889